1. Technical Field
The invention relates generally to measurement devices and more particularly to systems and methods for improving the accuracy of mass flow measurement and resulting control of fluids by correcting for physical differences between similar measurement devices.
2. Related Art
Thermal mass flow controllers (“MFCs”) measure the flow of a gas by sensing a temperature difference upstream and downstream of a heated section of capillary tubing through which the gas flows. The temperature difference between the upstream and downstream sensors is directly proportional (to the first order) to the gas' specific heat, or heat capacity, and the gas flow.
Thermal MFCs are typically calibrated with one gas and used with another, different gas. A different calibration gas is used because there are approximately 200 pure gasses and 300 mixtures that are used in the semiconductor industry. It would be impractical to calibrate an MFC with all of these gasses for a number of reasons. For one thing, the sheer number of gasses is very large. This problem is compounded by the fact that a manufacturer may have several different types of MFCs, each of which would have to be calibrated with each of the gasses. Another problem is that some of the gasses are corrosive, so it is preferable not to use them to calibrate the MFC. The MFC is therefore typically calibrated with a gas (preferably inert) which mimics the process gas. For example, calibration for a heavy process gas such as HCl3 may be performed with a heavy calibration gas such as SF6.
It should also be noted that the characteristics of the gasses may be different, so the same temperature difference between the upstream and downstream sensors may correspond to different flow rates of the gasses. The flow difference between the different gases is estimated using a gas correction factor. The gas correction factor is typically calculated by neglecting the effects of fluid dynamics, and relying upon the basic linearity of the device. Thus, the gas correction factor is normally just the ratio of the heat capacities of the process gas and calibration gas. In the prior art, a single gas correction factor (a single, constant value) is universally used for all models of MFCs and all flow ranges for a particular process gas. One manufacturer has used the same gas correction factors for over twenty years with a specified accuracy +/−5%.
Over the years, the accuracy of particular gas correction factors has occasionally been challenged. This has precipitated re-determination of the appropriate values for these particular correction factors. Even though the accuracy of the gas correction factors may be periodically re-confirmed, the overall accuracy of the measurement devices is still limited by the assumption of linearity.
Because the semiconductor industry is maturing and process control demands are increasing, more emphasis is being placed on the accuracy of process control instrumentation, including mass flow controllers. While the simple model for estimation of gas correction factors based on the heat capacity of the gas works well for the majority of semiconductor gases, a few gases which have been tested have non-linearities (relative to nitrogen) in excess of 5%. It is preferred to limit the nonlinearities to less than 5%.
For example, one tool may use a BCl3 MFC with a full scale of 200 sccm (standard centimeter cube per minute). The 200 sccm BCl3 device has a nitrogen equivalent flow of 489 sccm. That is, a 200 sccm flow of BCl3 produces the same sensor output as a 489 sccm flow of nitrogen. The nonlinearity of the 200 sccm BCl3 device is shown in FIG. 1. This figure shows the error in the flow measurement as a function of flow if a constant gas correction factor is used relative to nitrogen gas. The error shown is the non-linearity of the gas correction factor.
For virtually all process gases, if the flow through the sensor is limited to less than 2 sccm (nitrogen equivalent), the nonlinearity is less than 5%. If the flow through the sensor is larger than 2 sccm, significant nonlinearities may exist. For the example shown in FIG. 1, the error of the sensor at a flow of 200 sccm is 3.9 sccm (nitrogen equivalent).
The origin of the non-linearity is the breakdown of the assumed relationship of the surrogate gas to the process gas. The point at which the non-linearity becomes significant is a function of two parameters: 1) the gas flow through the sensor; and 2) the gas properties (specifically the ratio of the thermal conductivity to the heat capacity). The non-linearity is due to the fact that the gas does not fully thermally develop within the MFC sensor. The elementary theory assumes that the flow is fully “thermally developed”. The ability of the gas to be fully developed thermally is a function of the gas properties and the gas flow through the sensor. The flow through the sensor is adjustable, but the gas properties are fixed. If the flow through the sensor is constant, gases which will have large nonlinearities can be identified through examination of the gas properties. This evaluation has been accomplished for virtually all etch and chemical vapor deposition gases. Some of the gases which will exhibit this problem have been identified and are shown in Table 1.
TABLE 1Gases with Large NonlinearityGasK/Cp (ratio, in relative units)WF61.97E−01HBr2.10E−01BCl32.26E−01Cl22.66E−01
The information in Table 1 has been confirmed experimentally. Gases with a low thermal conductivity to heat capacity ratio will have difficulty achieving fully developed thermal profiles. In Table 1 the most nonlinear gas will be WF6, followed by HBr. This is consistent with the experimental data.
Because some semiconductor processing gases exhibit large nonlinearities, attempts have been made to compensate for the nonlinearity with the electronics of the MFCs. In existing MFC designs, this has been accomplished through the use of a correction factor that is a function of the flow rate of the gas being used. The gas correction factor (relationship between calibration gas and the process gas) is typically given by a function such asCF=CFo(1+aF+bF2+cF3)where CFo is the flow independent gas correction factor (often termed the “Gas correction factor”), F is the flow of the gas, and a, b, and c are gas-specific empirical or theoretical coefficients. This type of equation is adaptable to different gases, as all of the terms can be gas specific. Additionally, a family of curves can be developed for different TMFC configurations so that this type of function can be adapted for different designs. The use of the flow-dependent gas correction factor has yielded typical process accuracies of +/−1%.
While 1% accuracies are typical when flow-dependent gas correction factors are used, the error in the accuracies of different MFCs vary as a result of their manufacturing tolerances, and the range of accuracies will form a generally bell-shaped distribution around the ideal. Thus, while 1% error may be typical, there will be many MFCs that have error exceeding 1%. Some of the devices will substantially exceed this typical error. To date, the industry has not been able to conquer this problem, and the attainable accuracy with process gases has been limited to greater than 1% of reading.
Referring to FIG. 2, a diagram illustrating the differences in measurements made by several MFCs of identical design is shown. Each of the curves represents the measurements made by a single MFC. The curves are different because the MFCs each have different physical parameters, all of which are within the manufacturing tolerances of the MFC design. It can be seen from the figure that most of the MFCs provide measurements that are very close to each other. One of the MFCs, however, deviates substantially from the others and produces much higher readings.